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%I #18 May 10 2017 23:45:47
%S 1,1,1,2,1,4,2,1,3,4,2,6,1,3,6,4,4,2,6,5,1,8,3,9,10,4,12,4,2,15,6,9,5,
%T 1,10,8,3,6,9,20,10,4,7,12,4,12,2,8,15,6,14,16,5,18,1,16,20,8,18,3,6,
%U 19,9,24,20,10,28,4,7,30,12,32,4,12,35,2,8,14
%N Least possible strictly positive product of the two initial terms of a Fibonacci-like sequence containing n.
%C A Fibonacci-like sequence f satisfies f(n+2) = f(n+1) + f(n), and is uniquely identified by its two initial terms f(0) and f(1); here we consider Fibonacci-like sequences with f(0) > 0 and f(1) > 0.
%C This sequence is part of a family of variations of A249783, where we minimize a function g of the initial terms of Fibonacci-like sequences containing n:
%C - A249783: g(f) = f(0) + f(1),
%C - a: g(f) = f(0) * f(1),
%C - A286326: g(f) = max(f(0), f(1)),
%C - A286327: g(f) = f(0)^2 + f(1)^2.
%C For any n>0, a(n) <= n (as the Fibonacci-like sequence with initial terms n and 1 contains n).
%C For any n>0, a(A000045(n)) = 1.
%C For any n>2, a(A000032(n)) = 2.
%H Rémy Sigrist, <a href="/A286321/b286321.txt">Table of n, a(n) for n = 1..10000</a>
%H Rémy Sigrist, <a href="/A286321/a286321.txt">C program for A286321</a>
%H Rémy Sigrist, <a href="/A286321/a286321.png">Scatterplot of the first 100000 terms</a>
%H Rémy Sigrist, <a href="/A286321/a286321.pdf">Illustration of the first terms</a>
%e See illustration of the first terms in Links section.
%t Table[Module[{a = 0, b = 1, s = {}}, While[a <= n, AppendTo[s, Flatten@ NestWhileList[{#2, #1 + #2} & @@ # &, {a, b}, Last@ # < n &]]; If[a + b >= n, a++; b = 1, b++]]; First@ DeleteCases[#, 0] &@ Union@ Map[Times @@ #[[1 ;; 2]] &, Select[s, MemberQ[#, n] &]]], {n, 78}] (* _Michael De Vlieger_, May 10 2017 *)
%Y Cf. A000032, A000045, A249783, A286326, A286327.
%K nonn,look
%O 1,4
%A _Rémy Sigrist_, May 07 2017
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